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Elementary Bialgebra Properties of Group Rings and Enveloping Rings: An Introduction to Hopf Algebras.
- Source :
- Communications in Algebra; May2014, Vol. 42 Issue 5, p2222-2253, 32p
- Publication Year :
- 2014
-
Abstract
- This is a slight extension of an expository paper I wrote a while ago as a supplement to my joint work with Declan Quinn on Burnside's theorem for Hopf algebras. It was never published, but may still be of interest to students and beginning researchers. LetKbe a field, and letAbe an algebra overK. Then the tensor productA ⊗ A = A ⊗KAis also aK-algebra, and it is quite possible that there exists an algebra homomorphism Δ:A → A ⊗ A. Such a map Δ is called a comultiplication, and the seemingly innocuous assumption on its existence providesAwith a good deal of additional structure. For example, using Δ, one can define a tensor product on the collection ofA-modules, and whenAand Δ satisfy some rather mild axioms, thenAis called a bialgebra. Classical examples of bialgebras include group ringsK[G] and Lie algebra enveloping ringsU(L). Indeed, most of this paper is devoted to a relatively self-contained study of some elementary bialgebra properties of these examples. Furthermore, Δ determines a convolution product on HomK(A,A), and this leads quite naturally to the definition of a Hopf algebra. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00927872
- Volume :
- 42
- Issue :
- 5
- Database :
- Complementary Index
- Journal :
- Communications in Algebra
- Publication Type :
- Academic Journal
- Accession number :
- 93803503
- Full Text :
- https://doi.org/10.1080/00927872.2012.753604